Étale cohomology, cofinite generation, and $p$-adic $L$-functions

[Cohomologie étale, engendrement cofini, et fonctions $L$ $p$-adiques]

de Jeu, Rob ; Navilarekallu, Tejaswi

Annales de l'Institut Fourier, Tome 65 (2015), p. 2331-2383 / Harvested from Numdam

Résumé

Soit $p$ un nombre premier. Nous étudions certains groupes de cohomologie étale à coefficients associés à une représentation d’Artin $p$-adique de groupe de Galois d’un corps des nombres $k$. Ces coefficients sont munis d’un tordu à la Tate modifié avec un indice $p$-adique. Ces groupes sont de type cofini, et nous déterminons la caractéristique d’Euler additive. Si $k$ est totalement réel et la représentation est paire, nous étudions la relation entre le comportement ou la valeur de la fonction $L$ $p$-adique en le point $ e $ de ce domaine et les groupes de cohomologie avec torsion $p$-adique $1-e$. Dans certains cas, ceci donne une preuve courte d’une conjecture de Coates et Lichtenbaum, et de la conjecture équivariante des nombres de Tamagawa pour les fonctions $L$ classiques. Pour $p=2$ nos résultats impliquant des fonctions $L$ $p$-adiques dépendent d’une conjecture de la théorie d’Iwasawa.

Abstract

Let $p$ be a prime number. We study certain étale cohomology groups with coefficients associated to a $p$-adic Artin representation of the Galois group of a number field $k$. These coefficients are equipped with a modified Tate twist involving a $p$-adic index. The groups are cofinitely generated, and we determine the additive Euler characteristic. If $k$ is totally real and the representation is even, we study the relation between the behaviour or the value of the $p$-adic $L$-function at the point $e$ in its domain, and the cohomology groups with $p$-adic twist $1-e$. In certain cases this gives short proofs of a conjecture by Coates and Lichtenbaum, and the equivariant Tamagawa number conjecture for classical $L$-functions. For $p=2$ our results involving $p$-adic $L$-functions depend on a conjecture in Iwasawa theory.

Détail

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/aif.2989
Classification:  11G40,  14F20,  11M41,  11S40,  14G10
Mots clés: corps de nombres, cohomologie étale, génération cofinie, caractéristique d’Euler, fonction $ L $ d’Artin

Bibliographie

[1] Barrett, J.; Burns, D. Annihilating Selmer modules, J. Reine Angew. Math., Tome 675 (2013), pp. 191-222 | MR 3021451 | Zbl 1276.11173

[2] Báyer, P.; Neukirch, J. On values of zeta functions and $l$-adic Euler characteristics, Invent. Math., Tome 50 (1978/79) no. 1, pp. 35-64 | MR 516603 | Zbl 0409.12018

[3] Besser, A.; Buckingham, P.; De Jeu, R.; Roblot, X.-F. On the $p$-adic Beilinson conjecture for number fields, Pure Appl. Math. Q., Tome 5 (2009) no. 1, pp. 375-434 | Article | MR 2520465 | Zbl 1192.19003

[4] Bloch, Spencer; Kato, Kazuya $L$-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I, Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 86 (1990), pp. 333-400 | MR 1086888 | Zbl 0768.14001

[5] Bourbaki, N. Éléments de mathématique. Fasc. XXXI. Algèbre commutative. Chapitre 7: Diviseurs, Hermann, Paris, Actualités Scientifiques et Industrielles, No. 1314 (1965), pp. iii+146 pp. (1 foldout) | Zbl 0141.03501

[6] Burns, D. On main conjectures in non-commutative Iwasawa theory and related conjectures, J. Reine Angew. Math., Tome 698 (2015), pp. 105-159 | Article | MR 3294653

[7] Burns, D.; Flach, M. Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math., Tome 6 (2001), p. 501-570 (electronic) | MR 1884523 | Zbl 1052.11077

[8] Burns, D.; Greither, C. On the equivariant Tamagawa number conjecture for Tate motives, Invent. Math., Tome 153 (2003) no. 2, pp. 303-359 | Article | MR 1992015 | Zbl 1142.11076

[9] Chinburg, T.; Kolster, M.; Pappas, G.; Snaith, V. Galois structure of $ K $-groups of rings of integers, $K$-Theory, Tome 14 (1998) no. 4, pp. 319-369 | MR 1641555 | Zbl 0943.11051

[10] Coates, J.; Lichtenbaum, S. On $l$-adic zeta functions, Ann. of Math. (2), Tome 98 (1973), pp. 498-550 | MR 330107 | Zbl 0279.12005

[11] Ferrero, B.; Washington, L. The Iwasawa invariant $\mu _{p}$ vanishes for abelian number fields, Ann. of Math. (2), Tome 109 (1979) no. 2, pp. 377-395 | Article | MR 528968 | Zbl 0443.12001

[12] Flach, Matthias Euler characteristics in relative $K$-groups, Bull. London Math. Soc., Tome 32 (2000) no. 3, pp. 272-284 | Article | MR 1750171 | Zbl 1017.19002

[13] Flach, Matthias The equivariant Tamagawa number conjecture: a survey, Stark’s conjectures: recent work and new directions, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 358 (2004), pp. 79-125 (With an appendix by C. Greither) | MR 2088713 | Zbl 1070.11025

[14] Flach, Matthias On the cyclotomic main conjecture for the prime 2, J. Reine Angew. Math., Tome 661 (2011), pp. 1-36 | MR 2863902 | Zbl 1242.11083

[15] Fontaine, J.-M. Valeurs spéciales des fonctions $L$ des motifs, Astérisque (1992) no. 206, pp. Exp. No. 751, 4, 205-249 (Séminaire Bourbaki, Vol. 1991/92) | Numdam | MR 1206069 | Zbl 0799.14006

[16] Greenberg, R. On $p$-adic $L$-functions and cyclotomic fields. II, Nagoya Math. J., Tome 67 (1977), pp. 139-158 | MR 444614 | Zbl 0373.12007

[17] Greenberg, R. On $p$-adic Artin $L$-functions, Nagoya Math. J., Tome 89 (1983), pp. 77-87 | MR 692344 | Zbl 0513.12012

[18] Huber, A.; Kings, G. Bloch-Kato conjecture and Main Conjecture of Iwasawa theory for Dirichlet characters, Duke Math. J., Tome 119 (2003) no. 3, pp. 393-464 | Article | MR 2002643 | Zbl 1044.11095

[19] Illusie, Luc Cohomologie $l$-adique et fonctions $L$, Springer-Verlag, Berlin-New York, Lecture Notes in Mathematics, Vol. 589 (1977), pp. xii+484 (Séminaire de Géometrie Algébrique du Bois-Marie 1965–1966 (SGA 5)) | MR 491704

[20] Iwasawa, K. On ${\mathbb{Z}}_{l}$-extensions of algebraic number fields, Ann. of Math. (2), Tome 98 (1973), pp. 246-326 | MR 349627 | Zbl 0285.12008

[21] Jannsen, Uwe Continuous étale cohomology, Math. Ann., Tome 280 (1988) no. 2, pp. 207-245 | Article | MR 929536 | Zbl 0649.14011

[22] Knudsen, F. F.; Mumford, D. The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand., Tome 39 (1976) no. 1, pp. 19-55 | MR 437541 | Zbl 0343.14008

[23] Lichtenbaum, S. On the values of zeta and $L$-functions. I, Ann. of Math. (2), Tome 96 (1972), pp. 338-360 | MR 360527 | Zbl 0251.12002

[24] Milne, J. S. Étale cohomology, Princeton University Press, Princeton, N.J. (1980), pp. xiii+323 | MR 559531 | Zbl 0433.14012

[25] Milne, J. S. Arithmetic duality theorems, Academic Press, Inc., Boston, MA, Perspectives in Mathematics, Tome 1 (1986), pp. x+421 | MR 881804 | Zbl 0613.14019

[26] Neukirch, J. Algebraic number theory, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 322 (1999), pp. xviii+571 (Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder) | MR 1697859 | Zbl 0956.11021

[27] Neukirch, J.; Schmidt, A.; Wingberg, K. Cohomology of number fields, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 323 (2008), pp. xvi+825 | MR 2392026 | Zbl 1136.11001

[28] Perrin-Riou, B. $p$-adic $L$-functions and $p$-adic representations, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, SMF/AMS Texts and Monographs, Tome 3 (2000), pp. xx+150 (Translated from the 1995 French original by Leila Schneps and revised by the author) | MR 1743508 | Zbl 0988.11055

[29] Ribet, K. Report on $p$-adic $L$-functions over totally real fields, Journées Arithmétiques de Luminy (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978), Soc. Math. France, Paris (Astérisque) Tome 61 (1979), pp. 177-192 | MR 556672

[30] Schmidt, A. On the relation between 2 and $\infty $ in Galois cohomology of number fields, Compositio Math., Tome 133 (2002) no. 3, pp. 267-288 | Article | MR 1930978 | Zbl 1021.11029

[31] Serre, J.-P. Linear representations of finite groups, Springer-Verlag, New York (1977), pp. x+170 (Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42) | MR 450380 | Zbl 0355.20006

[32] Tate, J. Duality theorems in Galois cohomology over number fields, Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm (1963), pp. 288-295 | MR 175892 | Zbl 0126.07002

[33] Tate, J. Relations between $ K_{2} $ and Galois cohomology, Invent. Math., Tome 36 (1976), pp. 257-274 | MR 429837 | Zbl 0359.12011

[34] Washington, L. Introduction to cyclotomic fields, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 83 (1997), pp. xiv+487 | MR 1421575 | Zbl 0484.12001

[35] Wiles, A. The Iwasawa conjecture for totally real fields, Ann. of Math. (2), Tome 131 (1990) no. 3, pp. 493-540 | MR 1053488 | Zbl 0719.11071